A Monomial matrix formalism to describe quantum many-body states

Maarten Van den Nest Max Planck Institute for Quantum Optics arXiv:1108.0531

Montreal, October 19 th 2011

Motivation

Generalizing the Pauli stabilizer formalism

The Pauli stabilizer formalism (PSF)

 The PSF describes joint eigenspaces of sets of commuting Pauli operators 

i

: 

i |



= |



i = 1, …, k

 Encompasses important many-body states/spaces: cluster states, GHZ states, toric code, …  E.g. 1D cluster state: 

i = Z i-1 X i Z i+1

 The PSF is used in virtually all subfields of QIT:  Quantum error-correction, one-way QC, classical simulations, entanglement purification, information-theoretic protocols, …

Aim of this work

 Why is PSF so successful?

   Stabilizer picture offers

efficient description

Interesting quantities can be

efficiently computed

from this description (e.g. local observables, entanglement entropy, …) More generally:

understand

manipulating their stabilizers properties of states by  What are disadvantages of PSF?

 

Small

class of states

Special properties

: entanglement maximal or zero, cannot occur as unique ground states of two-local hamiltonians, commuting stabilizers, (often) zero correlation length… 

Aim of this work

: Generalize PSF by using larger class of stabilizer operators + keep pros and get rid of cons….

Outline

I.

II.

III.

IV.

Monomial stabilizers: definitions + examples Main characterizations Computational complexity & efficiency Outlook and conclusions

I. Monomial stabilizers

Definitions + examples

M-states/spaces

 Observation: Pauli operators are

monomial unitary matrices X =

 

0 1 1 0

 

Y =

 

0 i -i 0

 

Z =

 

1 0 0 -1

    Precisely one nonzero entry per row/column Nonzero entries are complex phases  M-state/space: arbitrary monomial unitary stabilizer operators U

i U i |

 

= |

 

i = 1, …, m

 Restrict to U i  with efficiently computable matrix elements E.g. k-local, poly-size quantum circuit of monomial operators, …

Examples

 M-states/spaces encompass many important state families:          All stabilizer states and codes (also for qudits) AKLT model Kitaev’s abelian + nonabelian quantum doubles W-states Dicke states Coherent probabilistic computations LME states (locally maximally entanglable) Coset states of abelian groups …

Example: AKLT model

  1D chain of spin-1 particles (open or periodic boundary conditions) H =  I-H i,i+1 where H i,i+1 is projector on subspace spanned by ψ 1  01  10 ψ 3  12  21 ψ 2  02  20 ψ 4  00  11  22   Ground level = zero energy: all | ψ  Consider monomial unitary

U :

with H i,i+1

|

ψ 

= |

ψ  01   10 12   21 02   20 00  11  22  00  Ground level = all | ψ  with U i,i+1

|

ψ 

= |

ψ  and thus

M-space

II. Main characterizations

How are properties of state/space reflected in properties of stabilizer group?

Notation:

computational basis |x  , |y 

, …

Two important groups

M-space

U i |

 

= |

 

i = 1, …, m



Stabilizer group

 = (finite) group generated by U

i



Permutation group

   Every monomial unitary matrix can be written as U = PD with P permutation matrix and D diagonal matrix. Call

U := P

Define 

:= {U : U

 

}

= group generated by U

i



Orbits:

 |y   O x O x = orbit of comp. basis state |x iff there exists U    and phase  under action of s.t. U|x  =  |y  

Characterizing M-states

 Consider M-state |ψ  and fix arbitrary |x  such that  ψ |x  

0



Claim 1:

All amplitudes are zero outside orbit O x : ψ ψ ψ x = | 1

G

|  U 

G

U x =  x 

Claim 2:

All nonzero amplitudes  y|ψ  have equal modulus   For all |y   Then  y| ψ  O

x

there exists U 

=

  x|U

* |

ψ  

=

  x| and phase  ψ  s.t. U|x  =  |y   Phase  is independent of U: 

=

 x (y)

M-states are uniform superpositions

   Fix arbitrary |x  such that  ψ |x  

0

All amplitudes are zero outside orbit O

x

All nonzero amplitudes have equal modulus with phase 

x (y) |

ψ  is

uniform superposition

over orbit ψ   x  Recipe to compute  x (y):  Find any U   such that s.t. U|x  =  |y  for some  ; then  =  x (y)  (Almost) complete characterization in terms of stabilizer group

Which orbit is the right one?

ψ   x  For every |x  let 

x

eigenvector. Then: be the subgroup of all U   which have |x  as O x is the correct orbit iff  x|U|x  = 1 for all U   x 

Example:

GHZ state with stabilizers Z i Z i+1 and X 1 …X n.

   O x 

x

= {|x  , |x + d generated by Z i Z i+1 Therefore O 0  = {|0  } where d = (1, …, 1) , |d for every x  } is correct orbit

M-spaces and the orbit basis

 Use similar ideas to construct basis of any M-space (orbit basis)

|

ψ 1 

|

ψ 2  B = {| ψ 1 

, … |

ψ d  }    Each basis state is

uniform superposition

These orbits are

disjoint

(  over some orbit dimension bounded by total # of orbits!) Phases  x (y) + “good” orbits can be computed analogous to before Computational basis

|

ψ d 

Example: AKLT model

(n even)

 Recall: monomial stabilizer for particles i and i+1 01   10 02   20 12   21 00  11  22  00  Generators of permutation group: replace +1 by -1  There are    

4 Orbits:

All basis states with even number of |0  s, |1  s and |2 

s

All basis states with odd number of |0  s and even number of |1  s, |2 

s

All basis states with odd number of |1  s and even number of |0  s, |2 

s

All basis states with odd number of |2  s and even number of |0  s, |1 

s

 Corollary: ground level

at most 4-fold

degenerate

Example: AKLT model

(n even)

01   10 02   20 12   21 00  11  22  00  Orbit basis for open boundary conditions:   a n  n σ = I,X, Y,Z 1 2  Unique ground state for periodic boundary conditions: ψ =   a n  n

III. Computational complexity and efficiency

NP hardness

 Consider an M-state | ψ  described in terms of diagonal unitary stabilizers acting on at most 3 qubits. 

Problem 1:

Compute (estimate) single-qubit reduced density operators (with some constant error) 

Problem 2:

Classically sample the distribution |  x|ψ  | 2  Both problems are

NP-hard

(Proof: reduction to 3SAT)  Under which conditions are efficient classical simulations possible?

Efficient classical simulations

 Consider M-state | ψ  Then |  x| ψ  | 2

can be sampled efficiently classically

following problems have efficient classical solutions: if the   Find an arbitrary |x  such that  ψ|x   0 Generate uniformly random element from the orbit of |x   Additional conditions to ensure that

local expectation values

can be estimated efficiently classically   Given y, does |x  belong to orbit of x? Given y in the orbit of x, compute  x (y)  Note: Simulations via

sampling

(weak simulations)

Efficient classical simulations

 Turns out: this general classical simulation method works for

all

examples given earlier         Pauli stabilizer states (also for qudits) AKLT model Kitaev’s abelian + nonabelian quantum doubles W-states Dicke states LME states (locally maximally entanglable) Coherent probabilistic computations Coset states of abelian groups  Yields

unified method

to simulate a number of state families

IV. Conclusions and outlook

Conclusions & Outlook

 Goal of this work was to demonstrate that: (1) (2) (3) M-states/spaces contain relevant state families, well beyond PSF Properties of M-states/-spaces can transparently be understood by manipulating their monomial stabilizer groups NP-hard in general but efficient classical simulations for interesting subclass  Many questions:      Construct new state families that can be treated with MSF 2D version of AKLT Connection to MPS/PEPS Physical meaning of monomiality …

Thank you!